Prime decomposition and correlation measure of finite quantum systems

Under the name prime decomposition (PD), a unique decomposition of an arbitrary N- dimensional density matrix rho into a sum of separable density matrices with dimensions determined by the coprime factors of N is introduced. For a class of density matrices a complete tensor product factorization is achieved. The construction is based on the Chinese remainder theorem. and the projective unitary representation of Z(N) by the discrete Heisenberg group H-N. The PD isomorphism is unitarily implemented and it is shown to be co-associative and to act on H-N as comultiplication. Density matrices with complete PD are interpreted as group-like elements of H-N. TO quantify the distance of rho from ifs PD a trace-norm correlation index epsilon is introduced and its invariance groups are determined.